We study the Goldbach problem for primes represented by the polynomial $x^{2}+y^{2}+1$ . The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^{2}+y^{2}+1$ . This improves a result of Matomäki, which tells us that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^{2}+y^{2}+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^{2}+y^{2}+1$ contain infinitely many three-term arithmetic progressions, and that the numbers $\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$ , with $\unicode[STIX]{x1D6FC}$ irrational and $p$ running through primes of the form $x^{2}+y^{2}+1$ , are distributed rather uniformly.

Let $s\geqslant 3$ be a fixed positive integer and let $a_{1},\ldots ,a_{s}\in \mathbb{Z}$ be arbitrary. We show that, on average over $k$ , the density of numbers represented by the degree $k$ diagonal form

$$\begin{eqnarray}a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}\end{eqnarray}$$

$k$

Recently, Freiman et al. [‘Small doubling in ordered groups’, J. Aust. Math. Soc. 96(3) (2014), 316–325] proved two ‘structure theorems’ for ordered groups. We give elementary proofs of these two theorems.

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$ , every sufficiently large odd integer $N$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$ , the interval $[p_{i},p_{i}+H]$ contains at least $m$ primes, for some $H=H(m)$ . Second, every sufficiently large integer $N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$ , $p_{i}+2$ has at most two prime factors.

A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$

$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$

$I_{s,k,r}(X)$

$1\leqslant r\leqslant k-1$

$s,k\in \mathbb{N}$

$k\geqslant 3$

$1\leqslant s\leqslant (k^{2}-1)/2$

$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$ , establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$ . In particular, as $k\rightarrow \infty$ , we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$ .

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$ -colored generalized Frobenius partitions of $n$ . Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$ .

Let $B_{k,\ell }(n)$ denote the number of $(k,\ell )$ -regular bipartitions of $n$ . Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of $B_{3,\ell }(n)$ and $B_{5,\ell }(n)$ . In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’, Ramanujan J.40, 535–540].

In this paper, we show that every pair of large positive even integers can be represented in the form of a pair of Goldbach–Linnik equations, that is, linear equations in two primes and $k$ powers of two. In particular, $k=34$ powers of two suffice, in general, and $k=18$ under the generalised Riemann hypothesis. Our result sharpens the number of powers of two in previous results, which gave $k=62$ , in general, and $k=31$ under the generalised Riemann hypothesis.

Consider a system of polynomials in many variables over the ring of integers of a number field  $K$ . We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_{K}^{m}$ satisfies the Hasse principle, weak approximation, and the Manin–Peyre conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$ . Our main tool is Skinner’s number field version of the Hardy–Littlewood circle method. As a by-product, we point out and correct an error in Skinner’s treatment of the singular integral.

Let G be an abelian group of cardinality n, where hcf(n, 6) = 1, and let A be a random subset of G. Form a graph ΓA on vertex set G by joining x to y if and only if x + y ∈ A. Then, with high probability as n → ∞, the chromatic number χ(ΓA ) is at most $(1 + o(1))\tfrac{n}{2\log_2 n}$ . This is asymptotically sharp when G = ℤ/nℤ, n prime.

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$ , and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$ -adic units in $\mathbb{Z}_{p}$ . It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime  $p$ .

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$ , there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$

$k=6$

$p$

$A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$

$g\in \{2,3,5\}$

$|A|\gg p/(\log p)^{3}$

$A\cdot A+A\cdot A$

$\mathbb{F}_{p}$

Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$, where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$. Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$. Finding the elements of the set ${\mathcal{A}}$ of the form $2^{k}m$, $k\geq 0$, is closely related to the $2$-adic integer $S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$, which has been shown to be an algebraic number. Let $G_{m}$ be the minimal polynomial of $S({\mathcal{A}},m)$. In precedent works there were treated the case $P$ irreducible of odd prime order $p$. In this setting, taking $p=1+ef$, where $f$ is the order of $2$ modulo $p$, explicit determinations of the coefficients of $G_{m}$ have been made for $e=2$ and 3. In this paper, we treat the case $e=4$ and use the cyclotomic numbers to make explicit $G_{m}$.

We revisit Berkovich and Garvan’s two bijections: the first gives symmetry of cranks and the second relates partitions with crank $\leq k$ to those with $k$ in the rank-set of partitions. Using these, we give a combinatorial proof for the relationship between the first positive crank moment and the sum of sizes of Durfee squares. We also study refinements of the first and second positive crank moments.

Let $b_{\ell }(n)$ denote the number of $\ell$-regular partitions of $n$. In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$-divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$. We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$.

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$ .

Let $E(N)$ denote the number of positive integers $n\leqslant N$ , with $n\equiv 4\;(\text{mod}\;24)$ , which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$ , thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$ .

In this paper, we investigate in various ways the representation of a large natural number as a sum of a fixed power of Piatetski-Shapiro numbers, thereby establishing a variant of the Hilbert–Waring problem with numbers from a sparse sequence.